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Vol. 51, 25–42 ©2022
http://doi.org/10.21711/231766362022/rmc512
Some remarks on an arbitrary-order SIR
model constructed with Mittag-Leffler
distribution
Noemi Zeraick Monteiro 1and Sandro Rodrigues
Mazorche 1
1Universidade Federal de Juiz de Fora, Juiz de Fora, Brazil
Abstract. Our recent works discuss the meaning of an arbitrary-
order SIR model. We claim that arbitrary-order derivatives can be
obtained through special power-laws in the infectivity and removal
functions. This work intends to summarize previous ideas and show
new results on a meaningful model constructed with Mittag-Leffler
functions. We emphasize the tricky idea to deal with equilibria, the
nonlocality of the model and the non-intuitive behavior near the
lower terminal.
Keywords: Fractional Calculus. Epidemiological model. Mittag-
Leffler functions. Nonlocality. Non-intuitive behaviors.
2020 Mathematics Subject Classification: 12A34, 67B89.
1 Introduction
“Mathematics is biology’s next microscope, only better; biology is math-
ematics’ next physics, only better” [1]. The quote by biomathematician
The first author is supported by the Coordination for the Improvement of Higher
Education Personnel (CAPES) - Brazil - Financing Code 001; e-mail of the correspond-
ing author: nzmonteiro@ice.ufjf.br.
25
26 N. Zeraick Monteiro and S. R. Mazorche
Joel E. Cohen portrays the current development of mathematical biology,
a branch that holds the attention of several researchers and has been on
the agenda every day during the current COVID-19 pandemic. The wide
use of mathematical models in situations related to biology does not ex-
haust their study. Contrariwise, some mathematical tools need a revision
to be used properly.
A great tool for problem modeling is Arbitrary-Order Calculus, known
as Fractional Calculus. In the most used definitions, there is the possi-
bility of explicitly considering the dependence of previous stages of the
phenomenon studied, through the nonlocality of the operators. This is
generally related to the “memory effect” [2]. The recent explosion of pub-
lications in Fractional Calculus highlights its immense applicability in nu-
merous areas (see, for instance, the data collected in [2]). However, the
basis is still not unified and coherent.
Compartmental models, for example, have been widely studied with
arbitrary orders. Generally, they are obtained by replacing an integer
derivative with an arbitrary-order one. Throughout our research, we pub-
lish some works about meaning difficulty, loss of properties and the lack
of the construction of fractional SIR type models ([3], [4], [5]). In this
context, we study a model proposed by Angstmann, Henry and McGann
[6] in which the arbitrary-order derivatives are obtained by construction,
considering Mittag-Leffler functions and generalizing the infectivity and
remotion functions. We seek to extend some analytical and numerical
results of the model in [4], [7], [8], [9].
Here, one of the aims is, after some preliminary presentations, dis-
cuss the idea around how to find equilibria in the proposed model, since
equating the right side of the system to zero is no longer a viable strategy.
The discussed strategy is somewhat trickier, using Laplace transform tech-
niques. After that, the main propose deals with two points that were not
discussed yet: as one would expect, the model is nonlocal and, moreover,
presents a non-intuitive behavior in the lower terminal.
Some remarks on an arbitrary-order SIR model 27
2 A brief note about nonlocality and the Frac-
tional Calculus
The impetus theory, studied by names as Leonardo Da Vinci, deals
with the concept of “impression”. According to Da Vinci, the impression is
maintained during a certain time in its sensitive object. This impression
(memory) characterized by a long time produces more lasting effects, while
a short memory produces effects that occur in shorter times [10]. Although
this reasoning was lost in the context of the Newtonian physics, in the last
century the quantum theory and the modern string theory allowed a revival
of nonlocal theories. Within that context, the Fractional Calculus can be
seen as a continuation or a sublimation of nonlocal concepts [11].
Broadly speaking, the Fractional Calculus, parallel to the delay differ-
ential equations, has been shown to be a very useful tool in capturing the
dynamics of the physical process of several scientific objects. Probably, it
was born in 1695, when l’Hôpital asked Leibniz about the meaning of a
derivative of order 1/2. Over the subsequent centuries, important advances
were made by Liouville, Riemann, Grünwald, Caputo, and many others.
However, it was only after the first International Conference on Fractional
Calculus and Applications, in 1974, that the number of researchers in Frac-
tional Calculus showed great growth. The reader may refer to the reference
[12] for a detailed chronology of publications in Fractional Calculus until
2019, as well as for general results.
Nonlocal operators can be constructed in different ways, depending on
the bias worked on. Here, before introducing the arbitrary-order integral,
we recall the concept of the integer-order integral, sometimes called the
multiple or iterated integral:
Definition 2.1 (Integer-order iterated integral).The integral of order
n∈Nis defined by the expression
Inf(t) = Zt
0Zt1
0Zt2
0
· · · Ztn−2
0Ztn−1
0
f(tn)dtndtn−1· · · dt3dt2dt1.(2.1)
By definition, I0f(t) = f(t).
28 N. Zeraick Monteiro and S. R. Mazorche
The next result, using the Laplace convolution, is a starting point for
the generalization of the concept of an integral of order n. For that, we
define:
Definition 2.2 (Gel’fand-Shilov function).Let α∈R,α > 0. The
Gel’fand-Shilov function is defined as
ϕα(t) =
tα−1
Γ(α)if t≥0,
0if t < 0,
(2.2)
where Γrepresents the gamma function.
Theorem 2.3. Let n∈N,0< t < ∞and f(t)be an integrable function.
Then,
Inf(t) = ϕn(t)⋆ f(t) = Zt
0
(t−τ)n−1
(n−1)! f(τ)dτ, (2.3)
where ⋆indicates convolution [13].
Thus, it is to be expected that the definition of an integral of arbitrary
order αis given by Iαf(t) = ϕα(t)⋆ f(t). Below, we consider [a, b]a finite
real interval, and αa real number such that 0≤n−1< α < n, with n
integer:
Definition 2.4 (Riemann-Liouville integral in finite intervals).The Riemann-
Liouville integral of an arbitrary order αis set to t∈[a, b]by
Iα
a+f(t) = 1
Γ(α)Zt
a
(t−θ)α−1f(θ)dθ. (2.4)
After introducing the arbitrary-order integral, it is natural to search
for the definition of the corresponding derivative. There are several defi-
nitions of these kind of derivatives, each one constructed with a particular
viewpoint. In this work, we use the Riemann-Liouville’s one:
Definition 2.5 (Riemann-Liouville derivative in finite intervals).The
Riemann-Liouville derivative of an arbitrary order αis set to t∈[a, b]
by
Dα
a+f(t) = Dn[In−α
a+f(t)] = 1
Γ(n−α)dn
dtnZt
a
(t−θ)n−α−1f(θ)dθ , (2.5)
Some remarks on an arbitrary-order SIR model 29
with Dnrepresenting the integer-order derivative.
Finally, we present the Mittag-Leffler functions with one, two, and
three parameters. The classic Mittag-Leffler function, due to its impor-
tance in several arbitrary-order differential equations, was nicknamed the
“queen of special functions” of the Fractional Calculus. Its importance
for Fractional Calculus is analogous to the significance of the exponential
function for classical Calculus. We present the following definition [12]:
Definition 2.6 (Mittag-Leffler function with one, two, and three param-
eters).Let zbe a complex number, and three parameters α, β complex,
and ρreal, such that Re(α)>0, Re(β)>0, ρ > 0. We define the Mittag-
Leffler function with three parameters through the power series
Eρ
α,β (z) =
∞
X
k=0
(ρ)k
Γ(αk +β)
zk
k!,(2.6)
where (ρ)kis the Pochhammer symbol, defined by (ρ)k= Γ(ρ+k)/Γ(ρ).
The three-parameter Mittag-Leffler function is also called Prabhakar
function. Particularly, when ρ= 1, we have (ρ)k=k!. In this case,
the definition recovers the two-parameter Mittag–Leffler function, denoted
simply by E1
α,β(t) = Eα,β (t). When ρ=β= 1, we obtain the classic
Mittag-Leffler function, denoted by E1
α,1(t) = Eα,1(t) = Eα(t). Finally, we
recover the exponential function when α=β=ρ= 1.
3 The model
We present in [4] a physical derivation following the steps of Angst-
mann, Henry & McGann [6], which use the probabilistic language of Con-
tinuous Time Random Walks (CTRW), and Mittag-Leffler functions. As
we can see with more detail in the references, the first idea is to consider
an individual infected since the time t′. If there are S(t)susceptible in
time t, this infected person has a probability S(t)/N that his contact is
susceptible, considering the population homogeneous. Therefore, in the
period of tto t+ ∆T, the expected number of new infections per infected
30 N. Zeraick Monteiro and S. R. Mazorche
individual is given by σ(t, t′)S(t)∆T/N. The transmission rate per infec-
tious individual σ(t, t′)depends on both the age of the infection, t−t′,
and the present time, t. The probability that an individual infected at the
moment t′is still infected at the moment tis given by the survival function
Φ(t, t′). Therefore, the flux of individuals to the Icompartment at a time
tis recursively given by
q+(I, t) = Zt
−∞
σ(t, t′)S(t)
NΦ(t, t′)q+(I, t′)dt′.(3.1)
To deal with the individuals infected at the time 0, we consider the
time in which each individual has become infected. This is given by the
function i(−t′,0) which represents the number of individuals who are still
infectious at time 0and who were originally infected at some point earlier
t′<0. Then, q+(I, t′) = i(−t′,0)/Φ(0, t′)for t′<0. For simplicity, we
consider i(−t, 0) = i0δ(−t), where δ(t)is the Dirac delta function. So,
q+(I, t) = Zt
0
σ(t, t′)S(t)
NΦ(t, t′)q+(I, t′)dt′+i0σ(t, 0)S(t)
NΦ(t, 0).(3.2)
As said, the infection rate σ(t, t′)is assumed to be a function of both
the current time (due, for example, to containment measures), having an
extrinsic infectivity, ω, and the age of infection, t−t′, having an intrinsic
infectivity, ρ. So, we can write
σ(t, t′) = ω(t)ρ(t−t′).(3.3)
Assuming that the natural death and the removal of an infected indi-
vidual are independent processes, we can write the survival function as
Φ(t, t′) = ϕ(t−t′)θ(t, t′),(3.4)
where ϕ(t−t′)is the probability that an individual infected since t′has not
yet recovered or been killed by the disease at time t. Also, θ(t, t′)is the
probability that an infected individual since t′has not yet died of natural
death (that is, independent of the disease) until time t. The θfunction is
given by θ(t, t′) = e−Rt
t′γ(u)du,where γis the death rate.
Some remarks on an arbitrary-order SIR model 31
We define infectivity and recovery memory kernels
KI(t) = L−1L{ρ(t)ϕ(t)}
L{ϕ(t)}, KR(t) = L−1L{ψ(t)}
L{ϕ(t)},(3.5)
where ψ(t) = −dϕ(t)/dt. This ψhas an important relationship with the
continuous random variable Xthat provides the time of removal of the
individual from the infectious compartment. The cumulative distribution
of X, namely Fdefined by F(t) = P(X≤t), is such that F(t) = 1 −ϕ(t).
Therefore, the probability density function of Xis ψ(t) = −dϕ(t)/dt. We
can state the set of equations for the SIR model in a similar manner to
that written originally by Kermack and McKendrick [14]:
dS(t)
dt =γ(t)N−ω(t)S(t)
Nθ(t, 0) Zt
0
KI(t−t′)I(t′)
θ(t′,0)dt′−γ(t)S(t),(3.6)
dI(t)
dt =ω(t)S(t)
Nθ(t, 0) Zt
0
KI(t−t′)I(t′)
θ(t′,0) dt′−θ(t, 0) Zt
0
KR(t−t′)I(t′)
θ(t′,0) dt′−γ(t)I(t),
(3.7)
dR(t)
dt =θ(t, 0) Zt
0
KR(t−t′)I(t′)
θ(t′,0)dt′−γ(t)R(t),(3.8)
where we consider the same rate γ(t)of natural mortality in each compart-
ment, with the birth rate equal to that. The population remains constant.
We choose ψ(t)and ρ(t)using Mittag-Leffler functions, in order to
generalize the exponential distribution of the random variable Xand allow
a variable intrinsic infectivity:
ϕ(t) = Eα,1−t
τα, ρ(t) = 1
ϕ(t)
tβ−1
τβEα,β −t
τα.(3.9)
Using Laplace transform techniques, the Riemann-Liouville derivatives
arise along the construction and the SIR model, with 1≥β≥α > 0, is
given by
dS(t)
dt =γ(t)N−ω(t)S(t)θ(t, 0)
Nτ βD1−βI(t)
θ(t, 0)−γ(t)S(t),(3.10)
dI(t)
dt =ω(t)S(t)θ(t, 0)
Nτ βD1−βI(t)
θ(t, 0) −θ(t, 0)
ταD1−αI(t)
θ(t, 0) −γ(t)I(t),(3.11)
dR(t)
dt =θ(t, 0)
ταD1−αI(t)
θ(t, 0)−γ(t)R(t),(3.12)
32 N. Zeraick Monteiro and S. R. Mazorche
Notice that, if α=β= 1, and γ(t)≡γ , ω(t)≡ωare considered
constant, we get the simple integer-order SIR model with constant coef-
ficients. Moreover, the cumulative distribution of Xis a Mittag-Leffler
distribution F(t;α, τ )=1−Eα(−(t/τ)α). If α=β= 1, we have an ex-
ponential distribution and the expectation (first moment) of the random
variable Xexists, with τbeing exactly the average recovery time. When
α < 1, we do not have finite expectation.
Remark 3.1. In epidemics such as COVID-19, reports exhibit the asym-
metry of each infectious wave: “while COVID-19 accelerates very fast, it
decelerates much more slowly. In other words, the way down is much
slower than the way up” [15]. In addition, scientists suggest that infected
people are most infectious immediately before they develop symptoms and
at the onset of them (e.g. [16]). The two factors presented, that is, the
asymmetry of the data, with a heavy right tail effect, and the decrease in
infectivity over time since infection, are captured by the arbitrary-order
model presented. As started in [4], [17], [18], we have been working on
applications of the model to COVID-19.
3.1 Equilibrium
Here, we analyze the equilibrium point (S∗, I∗, R∗)such that lim
t→∞(S, I, R) =
(S∗, I∗, R∗), where the limit is taken coordinate by coordinate. We con-
sider γ(t)≡γconstant, so θ(t, 0) = e−γt. Taking the limit t→ ∞, the
model reduces to:
0 = γN −lim
t→∞ ω(t)S(t)e−γt
Nτ βD1−β(I(t)eγ t)−γS∗,(3.13)
0 = lim
t→∞ ω(t)S(t)e−γt
Nτ βD1−β(I(t)eγ t)−e−γ t
ταD1−α(I(t)eγt )−γI ∗,(3.14)
0 = lim
t→∞ e−γt
ταD1−α(I(t)eγt )−γR∗.(3.15)
To calculate the limits of the form lim
t→∞ e−γt D1−α(I(t)eγt ), considering
γ > 0, we follow [19] and take the Laplace Transform:
L{e−γt D1−α(I(t)eγt )}= (s+γ)1−αL{I}.(3.16)
Some remarks on an arbitrary-order SIR model 33
Using a Taylor series expansion, we get
(s+γ)1−αL{I}=L{I}(γ1−α+ (1 −α)γ−αs+O(s2)).(3.17)
As the Laplace Transform is a linear operator, we can invert term by
term, obtaining
e−γt D1−α(I(t)eγt ) = γ1−αI(t) + (1 −α)γ−αdI
dt +L−1(O(s2)).(3.18)
As we consider lim
t→∞ I(t) = I∗, we have
lim
t→∞ dI/dt = lim
t→∞ L−1(O(s2)) = 0.(3.19)
So, it follows that
lim
t→∞ e−γt D1−α(I(t)eγt ) = γ1−αI∗.(3.20)
Substituting these results into Eq. (3.13)-(3.15) and assuming that
lim
t→∞ ω(t) = ω∗(possibly ω∗= 0), we are left with
0 = γN −ω∗S∗
Nτ βγ1−βI∗−γS∗,(3.21)
0 = ω∗S∗
Nτ βγ1−βI∗−1
ταγ1−αI∗−γI ∗,(3.22)
0 = 1
ταγ1−αI∗−γR∗.(3.23)
These equations make it possible to obtain a disease-free state:
S∗=N, I ∗= 0, R∗= 0,(3.24)
and, in the case where ω∗>0, we also obtain an endemic state:
S∗=((τ γ)β−α+ (τ γ )β)N
ω∗, I∗=N(τ γ )α
1+(τ γ)α−N(τ γ )β
ω∗, R∗=N
1+(τ γ)α−N(τ γ )β−α
ω∗.(3.25)
When ω(t)≡ω, we get ω∗=ωand recover the endemic state of the
original article [6]. We observe that the endemic state makes physical sense
only if we can have I∗>0and R∗>0, that is, if
ω∗>(τγ)β−α+ (τγ)β.(3.26)
34 N. Zeraick Monteiro and S. R. Mazorche
We have thus proven that, if there are asymptotically stable equilibria
in the case γ > 0, then they are given by Eq. (3.24)-(3.25). However,
we have not really proved that these states are asymptotically stable equi-
libria. We expect the disease-free state to be an asymptotically stable
equilibrium when ω∗<(τγ)β−α+ (τ γ)β, while the endemic state must be
asymptotically stable if ω∗>(τγ)β−α+ (τ γ)β.
For the case without vital dynamics, that is, γ= 0, if α=βand ω(t)≡
ω, we can write dS/dR =−(ωS(t)/Nτ), as in the original model, also
discussed in [4]. Thus, the equilibrium point is the same as in the original
case, as we state in [7]. For the case with γ > 0, there are difficulties to
analyze formally the stability. Some advances were also reported in [7].
4 Main remarks
There are several numerical methods that can be applied to arbitrary-
order derivatives. For now, we build a numerical L1-scheme [20] to dis-
cretize the model described in the last Section. The time interval [a, t]
is discretized as a=t0< t1<· · · < tn=t, where the time steps
∆Ti=ti+1 −ti, for i∈ {0,· · · , n −1}, have the same size ∆T. Consid-
ering α∈(0,1],we perform the following discretization for the Riemann-
Liouville derivative:
D1−α
a+f(tj)≃∆Tα−1
Γ(α+ 1) j−1
X
k=0
f(tk)[(j−k+ 1)α−2(j−k)α+ (j−k−1)α] + f(tj),(4.1)
where ti=i∆T+t0for all i∈ {0,1,· · · , n}. It is important to state that
the integer-order case is obtained by taking α= 1.
4.1 Nonlocality
We illustrate that the model (3.10)-(3.12) is nonlocal, as expected. For
this, we consider N= 106and initial conditions S(0) = N−1,I(0) =
1,R(0) = 0. At time t= 90, the numerical solution gives S(90) =
205890,I(90) = 267840, and R(90) = 526270. We now consider this
initial condition and run the model again.
Some remarks on an arbitrary-order SIR model 35
The Figure 4.1 illustrates the change of the solution, where the dashed
line corresponds to the solution from the time t= 90. In Figure 4.2,
we have the equivalent trajectories for a maximum time T= 3000. The
equilibrium is maintained.
0 50 100 150 200 250 300
t in days
0
2
4
6
8
10
S (blue), I (red), R(black)
#105
Parameters: !=3; ==10; .=0.01
,=0.7; -=0.9
Figure 4.1: Change in the solution.
0246810
S#105
0
0.5
1
1.5
2
2.5
3
I
#105
Parameters: !=3; ==10; .=0.01
,=0.7; -=0.9
Figure 4.2: Change in the trajectory.
Remark 4.1. In the integer-order SIR model, the epidemiological mean-
ingful parameters define the epidemic independently of time. Given an
initial condition (S(0), I(0), R(0)), and the solution (S(t), I (t), R(t)) of
the classic SIR model, let t∗>0: if we start at time t∗, with initial condi-
tion (S(t∗), I(t∗), R(t∗)), the solution will continue to be (S(t), I(t), R(t))
for t > t∗. This property of autonomous dynamical systems is called
invariance of solutions [21]. However, this is not valid here, making it
difficult to correctly choose initial conditions: what about the past prior
to the considered initial point? In this model, to adjust to the infection,
the parameters depend on the time series prior to the starting point. In
other words, different pasts will lead to different solutions in the future.
Starting a modeling in the second month of an epidemic, for example, does
not lead to the same result as when we start on the first day. It is worth
to mention that, to obtain the Equation (3.2), we used the Dirac delta
function to indicate that the disease does not exist before the time 0.
The nonlocality also implies an unexpected behavior of the reproduc-
tion number at time t′. It can be understood as the expected number
of individuals that are infected by an infectious individual since t′, with
36 N. Zeraick Monteiro and S. R. Mazorche
the basic reproduction number being the average number of secondary
infections that occur when an infectious individual is introduced into a
completely susceptible population [22]. In the constructed model, it is not
natural to define a formula that provides the reproduction number. Then,
in [6] the authors propose a construction for the basic reproduction num-
ber through an integral. Here, we extend the proposal of the reproduction
number for any time t′. Thus, initially, we consider ω(t)constant and the
definitions
ℜ0=ωγα−β
τβγα+τβ−α,(4.2)
ℜ(t′) = S(t′)ω
Nτ β
γα−β
γα+τ−α=ℜ0·S(t′)
N.(4.3)
This analysis is important, since for this model we do not have param-
eter tables, nor even prior application to any specific infectious disease.
As we can see here, in classical epidemiological models, the peak occurs
when ℜ(t)=1, but this does not occur in models of arbitrary orders or
with time-dependent parameters.
We consider the example below, in Figure 4.3, where the value of ℜ(t′)
given by Eq. (4.3) for the peak point t′is ℜpeak ≈0.9078 <1. If we
start modeling at a time before the peak, for which ℜpeak <ℜ(t)<1,
the epidemic will decrease even faster, but after a small rise. Note that
α=β. This will be discussed later. Also, is also possible to have no more
peaks if we start in a previous point, as we can see in Figure 4.4. In both
examples, we consider N= 106and the initial condition is (N−1,1,0).
We also can observe ℜpeak >1. In some of these cases, we can take an
earlier day, where ℜ(t)>ℜpeak >1, and the disease starts decreasing even
faster, as illustrated in Figure 4.5. In other words, “the point where the
epidemic must start to decrease” is really not the peak point anymore! This
indicates a difficulty in understanding the ℜ0defined in (4.2): although
ℜ0>1is the condition of endemic equilibrium viability given in (3.25),
ℜ0S(0)/N > 1does not necessarily indicate that the epidemic will or will
not occur, that is, whether or not Iwill grow.
Some remarks on an arbitrary-order SIR model 37
56 58 60 62 64 66 68 70
t in days
1.36
1.38
1.4
1.42
1.44
1.46
I
#105
Parameters: !=2; ==4; .=0.01
,=0.5; -=0.5
<(p-1)=0.9455
<(p)=0.9290
Figure 4.3: Case ℜ(p)<1(α=β).
100 105 110 115 120 125 130 135
t in days
1.55
1.6
1.65
1.7
1.75
I
#105
Parameters: !=2; ==7; .=0.01
,=0.5; -=0.6
<(p-1)=0.9878
<(p)=0.9750
Figure 4.4: Case ℜ(p)<1.
In this same Figure 4.5, we consider different starting points along
the originally constructed Icurve with initial condition (N−1,1,0). So,
it illustrates the evolution of the Icompartment if we start modeling at
different starting points along the same curve. The nonlocal behavior of
the model is surprising and we see that, in this example, the Icompartment
only rises if we take a starting point much earlier than the peak. Here, we
also start to consider the proposal that we presented in [7] for a S-variable
reproduction number:
ℜS(t′) = Z∞
t′
S(t)
Nτ βω(t)e−γ(t−t′)(t−t′)β−1Eα,β −(t−t′)
ταdt. (4.4)
In this proposal, the basic reproduction number is given by
ℜS
0=Z∞
0
S(t)
Nτ βω(t)e−γ ttβ−1Eα,β −t
ταdt. (4.5)
In the Figure table 4.6, we display the ℜ0S(0)/N and the ℜS
0S(0)/N
for each curve. In classical models, the disease declines since the beginning
if and only if ℜ0S(0)/N < 1. If ℜ0S(0)/N > 1, it must grow for a while.
As we can see, this does not happen with this model.
However, the equilibrium I∗given in (3.25) is the same for any initial
condition. So, broadly speaking, two simulations of the same disease with
different starting points lead to different solutions, but to the same end,
what is really important to study, for example, how many people will be
infected in total. Here we pointed out that, as will be explored in next
38 N. Zeraick Monteiro and S. R. Mazorche
25 30 35 40 45 50 55 60
t in days
1
1.5
2
2.5
3
3.5
4
4.5
I
#105
Parameters: !=5; ==9; .=0.01
,=0.5; -=0.9
Figure 4.5: Modifying the start point. Figure 4.6: ℜ0and ℜS
0.
0 100 200 300 400 500
t in days
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
I
#105
Parameters: !=5; ==9; .=0.01
,=0.5; -=0.9
Figure 4.7: The equilibrium remains the same.
works, the S−variable reproduction number seems to provide something
closer to what we expect, in terms of plausible biological arguments.
4.2 Behavior near lower terminal
Finally, we exhibit an unusual feature observed in the initial instants
of the model simulations. One can easily see in the Figure 4.5 that the
Some remarks on an arbitrary-order SIR model 39
blue and purple curves have a small depression before a rise. This type of
behavior does not appear in the integer-order model.
In fact, this can be explained given the asymptotic behavior of the
Riemann-Liouville derivatives near the lower terminal [23]:
Dα
a+f(t)≈f(a)
Γ(1 −α)(t−a)−α,(t→a+).(4.6)
So, in the model (3.10)-(3.12), if β > α, we have dI/dt < 0for tsmall
enough, as illustrated in the Figures 4.8-4.9.
0 0.2 0.4 0.6 0.8 1
t in days
0.7
0.8
0.9
1
1.1
1.2
1.3
I
Parameters: !=3; ==10; .=0.01
,=0.2:0.1:0.9; -=0.9
Figure 4.8: Lower terminal - Ex.1.
0 0.1 0.2 0.3 0.4
t in days
1
1.2
1.4
1.6
1.8
2
2.2
I
Parameters: !=3; ==10; .=0.01
,=0.4; -=0.4:0.1:0.9
Figure 4.9: Lower terminal - Ex.2.
When α=β, this behavior is not observed. For instance, we can see
in the pink curve of the Figure 4.3 that the behavior in the lower terminal
is the traditional movement of rising to a peak and decreasing after that.
Here, we also emphasize that the model assumes that there was no
disease before time 0, and then, at that point, an impulse caused the
disease to start. So, the historic is theoretically considered completely,
avoiding dubiousness about the starting point. However, the solution is not
C1, as it is not derivable at the origin. These remarks aim to strengthening
our intuition about the model and provide insights about its application.
5 Final considerations
So far, we have not been able to find a physical structure that allows
us to change the order of the original derivatives, even if the units are
40 N. Zeraick Monteiro and S. R. Mazorche
adjusted and the new orders are the same in all compartments. Therefore,
we seek the possibility of physically modeling a system, with a formalism
similar to that of Kermack and McKendrick, the creators of the SIR model.
In this sense, we have been discussing the almost unknown model of [6].
Here, we propagate the technique to work with equilibria in this type
of model and present important considerations, not previously explored:
the model is nonlocal, presents a difficult regarding the supposed relation
between the course of the epidemic and the reproduction number and, fi-
nally, there is an unexpected behavior at the initial point. Particularly,
start the simulation of an epidemic in its beginning or be able to say about
the past is a trick question. We aim that the deeper study of the equilib-
rium points and trajectories are fundamental predicting important features
of the model’s application. By other hand, the mathematics of the Frac-
tional Calculus’ models is still a black box of surprises that the assembling
between analytical and numerical studies can help to investigate.
Future works intend to close some results not fully explored, such as the
stability of equilibrium points in the general case and the use of an extrinsic
infectivity arising from Mittag-Leffler functions. The proposal of a variable
extrinsic infectivity was first illustrated in [4] and made it possible to fine-
tune the COVID-19 data, which leads us to new opportunities.
Acknowledgements
This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. We also thank
the team of the National Meeting of Mathematical Analysis and Applications
(ENAMA 2021), for the invitation to submit.
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